Conjecture on period lengths of square roots of primes

Let $p_{m}$ denote the $m$-th prime, let $T_{p_{m}}$ be the period length of the continued fraction expansion of $\sqrt{p_{m}}$, and define

$$\mathcal{W}_{i}:=\{p:T_{p}=i\}.$$

Period-length distribution conjecture. For each $i\in\mathbb{N}_{+}$, the limits

$$\lim_{N\rightarrow +\infty}\frac{\mathcal{W}_{2i+2}\cap\{p_{1},\ldots,p_{N}\}}{\mathcal{W}_{2i}\cap\{p_{1},\ldots,p_{N}\}},\quad \lim_{N\rightarrow +\infty}\frac{\mathcal{W}_{2i+1}\cap\{p_{1},\ldots,p_{N}\}}{\mathcal{W}_{2i-1}\cap\{p_{1},\ldots,p_{N}\}}$$

exist and both are equal to $1$. This conjecture is motivated by numerical data for primes up to the stated computational range; the source provides no proof or resolution.