Conjecture on the inverse matrix of the degenerate LGF $L_f(1,0,2,1;q)$

Let $p(n)$ denote the partition function, and let $s_{n,k}^{-1}(1,0,2,1)$ denote the entries of the inverse factorization matrix associated with the degenerate Lambert generating function $L_f(1,0,2,1;q)$. For integers $1\leq k\leq n$, define the Iverson bracket $[P]_{\delta}$ to be $1$ when the statement $P$ holds and $0$ otherwise.

Inverse-matrix conjecture. For $1\leq k\leq n$, the entries satisfy

$$s_{n,k}^{-1}(1,0,2,1)=p(n-k)-\sum_{i=1}^n p\left(\frac{n-i}{2i+1}-k\right)[n\equiv i\bmod 2i+1]_{\delta}$$ $$\qquad+\sum_{m=2}^n\sum_{i=1}^n p\left(\frac{n-p(m+1)i-p(m-1)}{p(m+1)(2i+1)}-k\right)[n\equiv p(m+1)i+p(m-1)\bmod p(m+1)(2i+1)]_{\delta}.$$

The claim is presented as a numerically observed property for this degenerate case; the source gives no proof or resolution, and generalizations for larger parameters are described only as preliminary computational observations.