Conjecture for even quadratic-polynomial values in the square-free real quadratic case

Let $d\equiv 2\pmod 4$ be a positive square-free integer. Define

$$M'_{\mathrm{even}}(d)=\max\{\omega(d-x^2):2\leq x\leq\sqrt d,\ x\text{ even}\}.$$

Conjecture. $M'_{\mathrm{even}}(d)\leq 2$ if and only if $d\in\{2,6,10,14,22,26,30,38,42,62,110,122,182,278,362,398\}$.

The claim concerns an unresolved real-quadratic analogue of the preceding factorization problems. The source says that it is suggested by easy numerical computations and that no finiteness result is currently available because the needed smallness of the fundamental unit has not been proved.