Nadeau–Tewari deletion–contraction conjecture for q-hit numbers

Let $\lambda$ be a partition inside an $n\times m$ board, let $e$ be an outer corner of $\lambda$, and let $H_j^{m,n}(\lambda)$ denote the associated $q$-hit number. Let $[m]_n$ denote the relevant $q$-falling-factorial quantity, and let $\lambda\backslash e$ and $\lambda/e$ be the partitions obtained by deleting and contracting $e$, respectively.

Nadeau–Tewari deletion–contraction conjecture. The $q$-hit numbers satisfy

$$H_j^{m,n}(\varnothing)=[m]_n\delta_{j,0},$$

and

$$qH_j^{m,n}(\lambda)=H_j^{m,n}(\lambda\backslash e)+q^mH_{j-1}^{m-1,n-1}(\lambda/e)-H_j^{m-1,n-1}(\lambda/e).$$

The recursion is a deletion–contraction relation intended to simplify the powers of $q$ in the existing recurrence for $q$-hit numbers. The source records that Nadeau–Tewari proved it in private communication.