Congruence relations among diagonal counts of Latin squares

Let $L$ be a Latin square of order $n$. For $1\leqslant k\leqslant n$, let $E_k$ be the number of diagonals of $L$ containing exactly $k$ distinct symbols. Let $t_{ij}$ be the number of transversals in the array obtained by deleting row $i$ and column $j$, and let $N_r$ and $R_k$ denote the row- and diagonal-count quantities defined in the paper. Congruence conjecture for diagonal counts. For all $i,j,r$, if $n\equiv0\pmod4$, then

$$E_n\equiv E_{n-1}\equiv 2E_{n-2}\equiv 2t_{ij}\equiv N_r\pmod4,$$ $$R_1+R_3+\cdots+R_{n-1}\equiv0\pmod4$$

and

$$R_2+R_4+\cdots+R_n\equiv E_n\pmod4.$$

If $n\equiv2\pmod4$, then

$$E_{n-1}\equiv2t_{ij}\equiv N_r\pmod4.$$

The conjecture extends the paper's established parity relations between consecutive diagonal counts, but its full modulo-$4$ assertions remain open.