We prove and generalize a conjecture in arXiv:1610.0474(4) about the asymptotics of $\frac{1}{\sqrt{n!}} f^{λ/μ}$, where $f^{λ/μ}$ is the number of standard Young tableaux of skew shape $λ/μ$ 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.
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