Parity conjecture for transversal types in even-order Latin squares

Let $L$ be a Latin square of even order $n$. Let $w$, $x$, $y$, and $z$ be the numbers of transversals in $L$ of types $T^{000}$, $T^{011}$, $T^{101}$, and $T^{110}$, respectively. Let $E_n(L)$ denote the number of diagonals of $L$ containing exactly $n$ distinct symbols, and let $E_n^{\pm}(L)$ denote the corresponding signed quantity used in the paper. Parity conjecture for transversal types.

$$E_n(L)\equiv E_n^{\pm}(L)\pmod 4$$

and

$$w\equiv x\equiv y\equiv z\pmod 2.$$

The statement is proposed on the basis of computational evidence as a generalisation of earlier results, and its general validity remains open.