Conjecture: $\sum_{m=2}^L \,\,\sum_{l=\max(M+m-L,1)}^{\min(m-1,M)}{L-m \choose M-l}={L \choose M}-1$

Question

I conjecture, and have verified on Mathematica, that $$\sum_{m=2}^L \,\,\sum_{l=\max(M+m-L,1)}^{\min(m-1,M)}{L-m \choose M-l}={L \choose M}-1\,.$$ How I arrived at this conjecture is a long-but-interesting story related to my quantum computing research.

I understand that the min/max are enforcing the requirement $0\leq M-l\leq L-m$, but it remains unclear to me how to prove this conjecture. Any ideas?