  We show that the metric defined by the solution to the tropical
Monge-Amp\`ere equation, as defined by Hultgren, Mazzon, and the first two
authors, on the boundary of the 3-simplex is asymptotic to the Gross-Wilson
metric on $S^2$ near each of the 6 singular points. We deduce in addition that
the solution is not $C^{1,1}$ across the singular points. Compared to previous
works, our starting point is the real Monge-Amp\`ere equation, as opposed to
the complex structure.
