Modified Max Product dynamic solution very inefficient
Why do you split it in the middle? Multiplication is associative so you don't have to do that. Considering only the last multiply sign would be fine.
Here's a simpler recursion: $T(1,j,a)=\max\{T(1,v,a-1)\times T(v+1,j,0)\}$. I use the same symbols as yours so you can see the difference. The first dimension is useless so the table is actually an $n \times k$ array. Each function call takes $O(n)$, so the complexity is $O(n^2k)$.