  We prove and generalize a conjecture in arXiv:1610.0474(4) about the
asymptotics of $\frac{1}{\sqrt{n!}} f^{\lambda/\mu}$, where $f^{\lambda/\mu}$
is the number of standard Young tableaux of skew shape $\lambda/\mu$ which have
stable limit shape under the $1/\sqrt{n}$ scaling. The proof is based on the
variational principle on the partition function of certain weighted lozenge
tilings.
