Povolotsky's recurrence conjecture for perfect lattice paths

Let $\mathcal{I}_n(n)$ denote the number of the relevant perfect lattice paths in an $n\times n$ table. Povolotsky's recurrence conjecture. For positive integers $n$, the following identity holds:

$$(n+3)\mathcal{I}_{n+4}(n+4)=27n\mathcal{I}_n(n)+27\mathcal{I}_{n+1}(n+1)-9(2n+5)\mathcal{I}_{n+2}(n+2)+(8n+2)\mathcal{I}_{n+3}(n+3).$$

The identity is presented as a conjecture of Alexander R. Povolotsky and is proved in the source using the preceding recurrence relations, so it is resolved rather than open.