Gamma-product evaluations for the E-determinant family

Let $E_{a,b,c,d}(n)$ denote the determinant family used in the source, and define

$$\Xi(x):=\prod_{i=2}^x\frac{3\Gamma(i)\Gamma(4i-3)\Gamma(4i-2)}{2\Gamma(3i-2)^2\Gamma(3i-1)},\qquad \mu_m(x):=\begin{cases}2,&3\mid(x-m),\\1,&\text{otherwise}.\end{cases}$$

Gamma-product conjecture for the E-determinants. For all non-negative integers $x$ and all $n\geq x$, the three displayed formulas in the source hold:

$$E_{0,x,-x,-3x}(n)=2\mu_1(x)\Xi(x)(-1)^{\lfloor x/3\rfloor}\prod_{i=1}^n\frac{2^{i-1}\Gamma(4i-3)\Gamma((i+1)/3)}{\Gamma(3i-2)\Gamma((4i-2)/3)},$$ $$E_{1,x,1-x,1-3x}(n)=2\mu_2(x)\Xi(x)(-1)^{\lfloor(x+2)/3\rfloor}\prod_{i=1}^n\frac{2^{i-2}\Gamma(4i-1)\Gamma(i/3)}{3\Gamma(3i-1)\Gamma(4i/3)},$$ $$E_{2,x,2-x,2-3x}(n)=\frac{\mu_0(x)}{n}\Xi(x)(-1)^{\lfloor(x+1)/3\rfloor}\prod_{i=2}^n\frac{2^{i-3}\Gamma(4i+1)\Gamma((i-1)/3)}{9\Gamma(3i)\Gamma((4i+2)/3)}.$$

These are closed-form determinant evaluations in a broader family of identities. The supplied material gives no proof or status evidence for this conjectural span.