I can verify that lma03's solution is correct and unique, using backtracking and a few very simple checks for forced moves. Here's a way to solve it (trying all deadends first)...

Level 35: location c7r9 -- multiple possibilities 1 8

Level 35: location c7r9 -- insert a 1

c7r7 -> 8 c9r2 -> 1 c8r8 -> 7 c8r3 -> 5 c6r7 -> 7 c7r5 -> 7 c1r2 -> 5 c9r5 -> 4 c2r6 -> 7 c2r2 -> 4 c1r6 -> 8 c3r1 -> 7 c2r4 -> 2

Level 49: deadend detected at location c3r7

Level 35: location c7r9 -- insert an 8

Level 36: location c8r3 -- multiple possibilities 3 5

Level 36: location c8r3 -- insert a 3

c8r4 -> 2 c9r2 -> 5 c6r2 -> 3 c2r4 -> 4 c8r6 -> 7 c8r8 -> 5 c6r6 -> 2 c5r5 -> 3 c5r1 -> 9 c1r4 -> 3 c7r7 -> 7 c3r5 -> 7

Level 49: deadend detected at location c1r5

Level 36: location c8r3 -- insert a 5

c1r2 -> 5

Level 38: location c8r4 -- multiple possibilities 2 3

Level 38: location c8r4 -- insert a 3

c8r8 -> 7 c8r6 -> 2 c9r5 -> 4 c7r5 -> 7 c6r7 -> 7 c6r6 -> 3 c5r5 -> 2 c2r6 -> 7 c1r6 -> 8 c5r3 -> 3 c6r9 -> 2 c3r1 -> 7 c2r7 -> 2 c4r2 -> 7 c3r3 -> 8

Level 54: deadend detected at location c3r9

Level 38: location c8r4 -- insert a 2

c2r4 -> 4 c1r4 -> 3

Level 41: location c8r8 -- multiple possibilities 3 7

Level 41: location c8r8 -- insert a 3

c8r6 -> 7 c7r7 -> 7 c3r7 -> 3 c3r5 -> 7 c2r2 -> 7 c1r5 -> 6 c4r2 -> 4 c4r1 -> 7 c3r3 -> 6 c4r8 -> 8 c6r8 -> 7 c2r7 -> 8 c6r9 -> 4 c5r8 -> 9 c2r9 -> 2

Level 57: deadend detected at location c1r6

Level 41: location c8r8 -- insert a 7

c8r6 -> 3 c7r5 -> 7 c6r7 -> 7 c6r6 -> 2 c5r5 -> 3 c9r5 -> 4 c3r5 -> 6 c2r6 -> 7 c1r5 -> 2 c1r6 -> 8 c1r3 -> 6 c2r2 -> 1 c3r1 -> 7 c2r9 -> 2 c9r2 -> 3 c1r8 -> 1 c1r1 -> 4 c4r2 -> 7 c3r3 -> 8 c2r7 -> 8 c5r7 -> 2 c7r7 -> 3 c6r3 -> 3 c9r8 -> 5 c4r1 -> 8 c5r3 -> 1 c9r7 -> 1 c3r7 -> 5 c3r8 -> 3 c6r2 -> 4 c5r9 -> 5 c4r8 -> 4 c5r8 -> 8 c6r9 -> 1 c7r3 -> 4 c7r1 -> 1 c5r1 -> 9 c3r9 -> 4 c7r2 -> 9 c6r8 -> 9

solved!

4 3 7 8 9 5 1 6 2

5 1 2 7 6 4 9 8 3

6 9 8 2 1 3 4 5 7

3 4 9 1 7 6 5 2 8

2 5 6 9 3 8 7 1 4

8 7 1 5 4 2 6 3 9

9 8 5 6 2 7 3 4 1

1 6 3 4 8 9 2 7 5

7 2 4 3 5 1 8 9 6