Strong coupling expansion of $1\over 2$ BPS Wilson loop in SYM theory and 2-loop Green-Schwarz string in AdS$_5 \times $S$^5$

The exact localization result for the expectation value of the $1\over 2$ BPS circular Wilson loop in ${\cal N}=4$ SYM theory is given in the planar limit by the famous Bessel function expression: $\langle W\rangle = {2N\over \sqrt \lambda } I_1 ( \sqrt \lambda)$. Expanded in large $\lambda$ and expressed in terms of the AdS$_5 \times$S$^5$ string tension $T= {\sqrt \lambda \over 2\pi}$ this gives $\langle W\rangle = {\sqrt T\over 2\pi g_s} e^{2\pi T} (1- {3\over 16 \pi} T^{-1} + ...)$.The exponential is matched by the value of the action of the string with the AdS$_2$ world volume while the prefactor comes from the 1-loop GS string correction. Here we address the question of how the subleading $T^{-1}$ term could be reproduced by the 2-loop correction in the corresponding partition function of the AdS$_5 \times$S$^5$ GS string expanded near the AdS$_2$ minimal surface. We find that the string correction contains a non-zero UV logarithmic divergence implying that comparison with the SYM result requires a particular subtraction prescription. We discuss implications of this conclusion for checking the AdS/CFT duality at strong coupling.