Catalan-number conjecture for 2-adic orders with two binary digits

Let $d_2(k)$ be the sum of the binary digits of $k$, and let $C_{1,n}$ be the $n$th positive Catalan number other than $C_0$ having 2-adic order $1$. Suppose that $k$ is the $n$th positive integer with $d_2(k)=2$, $m=4j$ for $j=1,2,\ldots$, $a=\operatorname{ord}_2(m)$, and $o=\operatorname{ord}_2(A_{\overline{\mathcal K},k,m}(0))$. Define

$$t=\frac{(a+6)k+2-o}{4}.$$

Catalan valuation conjecture. Then $t=C_{1,n}$. The source presents this as an observed pattern in its empirical study of constant-term valuations; no resolution is given.