.---------------------------.
7 | 16  __  __  __  __  __  49|
6 | __  19  __  __  __  __  __|
5 | __  27  __  __  __  __  __|
4 | __  24  __  __  __  __  10|
3 | 29  __  __  __  __  __  __|
2 | __  __  __  34  39  __  42|
1 | __  __  35  __  __  __  01|
  '---------------------------'
    a   b   c   d   e   f   g
    

Update 1 (Sept 01, 2026)

The positions 10 and 16 makes
possible to get 4 numbers 
(marked with *).

  .---------------------------.
7 | 16 *15  __  __  __  __  49|
6 |*17  19  __  __  __  __  __|
5 |*18  27  __  __  __  __  __|
4 |*28  24  __  __  __  __  10|
3 | 29  __  __  __  __  __  __|
2 | __  __  __  34  39  __  42|
1 | __  __  35  __  __  __  01|
  '---------------------------'
    a   b   c   d   e   f   g

How to advance from here ?
    

Update 2 (Sept 3, 2026)
    
After 4 numbers, a huge barrier seems
to exist for the next one! We have to
look for weaknesses and the natural
place for that is the lower right
corner. I have found a strong but
very complex move:

       (40)f3 -> (43)f2, (44)e3, (38)d3 and path 01-10 leaves no place for 33
      /
(41)g3
 ||   \
 ||    (40)f2 -> path 2-10, locked numbers (06,33,38)c2.d2.e2 and path 35->39 are in conflict
 ||
 ||    (02)f2 -> (40)e3, (43)g3!, and path 2-10 is tight -> no place for 38
 ||   /
(41)f3        (43)f2 -> (40)e3, (44)g3, (45)f4, (38)d3!, and path 2-10 leaves no place for 33
 ||   \      /
 ||    (02)f1       (3)f2 -> (40)e3, and path 3-10 leaves place for 38
 ||          \     /
 ||          (43)g3
 ||                \
 ||                 (3)e1!!,(4)d1!,(5)c2, (44)f4!, now path 35-39 leaves no place for 33
 || 
(41)f2->(02)f1, (03)e1, (04)d1, (05)c2, (36)b2!, (37)c3,(38)d3, (33)e3, and path 5->10 is impossible.

Since those three possible places for 41 each gives a contradiction (each as described), we must have

5.(41)f1!!! [three due to the complexity, but I would happier with a simpler move]

6.(02)f2 7.(40)e1 8.(38)d1 9.(37)c2

So, we get

  .---------------------------.
7 | 16 *15  __  __  __  __  49|
6 |*17  19  __  __  __  __  __|
5 |*18  27  __  __  __  __  __|
4 |*28  24  __  __  __  __  10|
3 | 29  __  __  __  __  __  __|
2 | __  __ *37  34  39 *02  42|
1 | __  __  35 *38 *40 *41  01|
  '---------------------------'
    a   b   c   d   e   f   g

The situation has improved a lot,
it seems. How to continue ?
    

Update 3 (Sept 4, 2026)

Now, small deductions get many
placements. The first one is
10.(36)b1! [(36)b2 leads to trouble
with that corner at b1]. After 7
more placements, we get

  .---------------------------.
7 | 16 *15  __  __  __  __  49|
6 |*17  19  __  __  __  __  __|
5 |*18  27 *23  __  __  __  __|
4 |*28  24 *26  __  __  __  10|
3 | 29 *25 *33  __  __  __  __|
2 |*30 *32 *37  34  39 *02  42|
1 |*31 *36  35 *38 *40 *41  01|
  '---------------------------'
    a   b   c   d   e   f   g

How to make progress now ?
    

Update 4 (Sept 6, 2026)
    
The following move gives only two numbers
but it contains the main ideas involved
in the resolution of this puzzle.

       (21)d6, (13)d7,(12)e6,(11)f5,(22)d5, and conflicting paths 02-10,42-49.
      /
(20)c6        (22)d6, (13)d7,(12)e6,(11)f5,(21)d5, and conflicting paths 02-10,42-49.
     \      /
      (21)d5
            \
              (22)d4, path 42-49 must pass through one of the cells e4, f4, and
              use at least one cell of line 3. The path 2-10 has 7 numbers, so
              it must use either e4 or f4. So, for the path 10-14, we must have
              (11)f5!, which narrows the ways for the other two paths. One of
              them must pass through f5, g5 and the other must use the cell e5.
              So, (12)e6! is forced. Now the only way for the path 2-10 is
              to use both passages, which makes the path 42-49 impossible.

------
The two next grids are for the analysis for the last part 
(the most interesting, since it involves conflict of three paths)

  .---------------------------.
7 | 16 *15 *14  __  __  __  49|
6 |*17  19 ?20  __  __  __  __|
5 |*18  27 *23 ?21  __  __  __|
4 |*28  24 *26 *22  __  __  10|
3 | 29 *25 *33  __  __  __  __| 03,43
2 |*30 *32 *37  34  39 *02  42|
1 |*31 *36  35 *38 *40 *41  01|
  '---------------------------'
    a   b   c   d   e   f   g

  .---------------------------.
7 | 16 *15 *14  __  __  __  49|
6 |*17  19 ?20  __ *12  __  __|
5 |*18  27 *23 ?21  __ *11  __|
4 |*28  24 *26 *22  __  __  10|
3 | 29 *25 *33  __  __  __  __| 03,43
2 |*30 *32 *37  34  39 *02  42|
1 |*31 *36  35 *38 *40 *41  01|
  '---------------------------'
    a   b   c   d   e   f   g

----
So, 18.(20)c7!!! 19.(14)c6

  .---------------------------.
7 | 16 *15 *20  __  __  __  49|
6 |*17  19 *14  __  __  __  __|
5 |*18  27 *23  __  __  __  __|
4 |*28  24 *26  __  __  __  10|
3 | 29 *25 *33  __  __  __  __|
2 |*30 *32 *37  34  39 *02  42|
1 |*31 *36  35 *38 *40 *41  01|
  '---------------------------'
    a   b   c   d   e   f   g

Still needs work, but a lot less than
the two previous moves.