The conjecture that iterated divisor-preimage minima are highly composite

Let $d(n)$ denote the divisor function, and let $d^k$ denote its $k$-fold iterate. For each $k$, let $n_k$ be the least integer such that $d^k(n_k)=2$. A highly composite number is a positive integer having more divisors than every smaller positive integer.

Highly composite minima conjecture. All the integers $n_k$ produced by Theorem 1 are highly composite numbers.

The observed initial values are $n_1=2$, $n_2=4$, $n_3=6$, $n_4=12$, $n_5=60$, and $n_6=5040$. The proposed heuristic uses the maximal-order estimate for the divisor function to suggest that $n_{k-1}$ is asymptotic to the largest value of $d(n)$ for $n\leq n_k$, which would imply the claimed highly composite property.