Antisymmetric Wilson loops in $N=4$ SYM beyond the planar limit

We study the $\frac{1}{2}$-BPS circular Wilson loop in the totally antisymmetric representation of the gauge group in $N=4$ supersymmetric Yang-Mills. This observable is captured by a Gaussian matrix model with appropriate insertion. We compute the first $1/N$ correction at leading order in 't Hooft coupling by means of the matrix model loop equations. Disagreement with the 1-loop effective action of the holographically dual $D5$-brane suggests the need to account for gravitational backreaction on the string theory side.