Congruence conjecture for permanents of the absolute-difference matrix

Let $p$ be an odd prime, let $\delta_{jk}$ be the Kronecker delta, and let $\operatorname{per}(M)$ denote the permanent of a matrix $M$. Absolute-difference permanent conjecture.

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

The conjecture is presented as being inspired by a related congruence for $\operatorname{per}[|j-k|]$ and by the paper's determinant evaluation. The source gives no evidence that this assertion has been resolved.