QTC2 plane partitions enumeration conjecture

Let $\pi$ be a plane partition inside an $(a,a,c)$-box. It is quasi transpose complementary of second kind (QTC2) if it is transpose complementary except along the diagonal, namely

$$\pi_{i,j}+\pi_{a+1-j,a+1-i}=c$$

for all $1\leq i,j\leq a$ with $i\neq j$. Let $\operatorname{qtcpp}_2(a,c)$ denote the number of QTC2 plane partitions, and let $p_a(c)$ be a polynomial in $\mathbb{Q}[c]$. QTC2 enumeration conjecture. For $a\geq2$,

$$\operatorname{qtcpp}_2\left(a,c-\frac{a}{2}\right)=\begin{cases} c\binom{c+\frac{a}{2}-1}{a-1}p_a(c)&a\text{ is even},\\ \binom{c+\frac{a}{2}-1}{a-1}p_a(c)&a\text{ is odd}. \end{cases}$$

Here $p_a(c)$ is irreducible and even, and the common denominator of its coefficients is a product of “small primes”. The formula was proposed from computer experiments and remains unproved.