Infinitely many Catalan numbers with proportional 2- and 3-adic valuations

Let $C_n=\frac{1}{n+1}\binom{2n}{n}$ denote the $n$th Catalan number, and let $\nu_p(m)$ be the exponent of the prime $p$ in the prime factorization of $m$. Assume that $a,b\geq 1$ are integers. Catalan valuation conjecture. There exist infinitely many positive integers $n$ such that

$$a\nu_2\bigl(C_n\bigr)=b\nu_3\bigl(C_n\bigr).$$

This is a related question about the 2- and 3-adic valuations of Catalan numbers, analogous to the paper's results on collisions between binary and ternary digit sums. The source leaves this as another open problem.