On the period of the continued fraction expansion for $\sqrt{d}$

If $d$ is not a perfect square, we define $T(d)$ as the length of the minimal period of the simple continued fraction expansion for $\sqrt{d}$. Otherwise, we put $T(d)=0$. In the recent paper (2024), F. Battistoni, L. Grenié and G. Molteni established (in particular) an upper bound for the second moment of $T(d)$ over the segment $x<d\leqslant 2x$. As a corollary, they derived a new upper estimate for the number of $d$ such that $T(d)>\alpha\sqrt{x}$. In this paper, we slightly improve this result of three authors.