Recent progress on the next-to-shortest-path problem for directed graphs? In the paper "Computing strictly-second shortest paths" (1997), Lalgudi and Papaefthymiou consider the following problem:

Let $G$ be a directed graph with edge-weighting $w$. Let $u,v$ be vertices in $V(G)$. Let $p_1$ be a path of minimum weight from $u$ to $v$. Compute a simple path $p_2$ from $u$ to $v$ with $w(p_2) &gt; w(p_1)$ and such that for any other path $p$ from $u$ to $v$ with $w(p) &gt; w(p_1)$, we have $w(p) \ge w(p_2)$. 

They prove that this problem is NP-hard. Their proof involves choosing a weight function that takes value $1$ on a certain edge and $0$ elsewhere, so it relies in a fundamental way on the possibility of there being edges of weight $0$.

Is anything known about whether the problem remains NP-hard when one restricts to unit-weight graphs, that is graphs for which $w(e) = 1$ for each $e \in E(G)$?
My first intuition is certainly naive. What's wrong with it?

Since $p_2$ is different from $p_1$ on at least one edge, repeatedly remove one edge in $p_1$ from the graph and solve the shortest path problem, which would be at worst $n$ times the complexity of the shortest path problem.