Congruence conjectures for permanents of shifted difference matrices

For a positive integer $n$, let $[j-k+\delta_{jk}]_{1\leqslant j,k\leqslant n}$ denote the $n\times n$ matrix whose $(j,k)$ entry is $j-k+\delta_{jk}$, where $\delta_{jk}$ is the Kronecker delta, and let $\operatorname{per}(M)$ denote the permanent of a matrix $M$. Congruence conjecture. For any prime $p$,

$$\operatorname{per}[j-k+\delta_{jk}]_{1\leqslant j,k\leqslant p-1}\equiv 3\pmod p.$$

For any odd prime $p$,

$$\operatorname{per}[j-k+\delta_{jk}]_{1\leqslant j,k\leqslant p}\equiv 1+4p\pmod {p^2}.$$

These assertions are motivated by the preceding determinant evaluation in the paper; no resolution or external evidence concerning either congruence is supplied here.