  We present the calculation of the light neutral CP-even Higgs mass in the
MSSM for a heavy SUSY spectrum by resumming enhanced terms through fourth
logarithmic order (N$^3$LL), keeping terms of leading order in the top Yukawa
coupling $\alpha_t$, and NNLO in the strong coupling $\alpha_s$. To this goal,
the three-loop matching coefficient for the quartic Higgs coupling of the SM to
the MSSM is derived to order $\alpha_t^2\alpha_s^2$ by comparing the
perturbative EFT to the fixed-order expression for the Higgs mass. The new
matching coefficient is made available through an updated version of the
program Himalaya. Numerical effects of the higher-order resummation are studied
using specific examples, and sources of theoretical uncertainty on this result
are discussed.
