Is there a gap amplification type of result for the Graph Isomorphism Problem? Suppose $G_1$ and $G_2$ are two undirected graphs on vertex set $\{1, \dotsc, n\}$. The graphs are isomorphic if and only if there is a permutation $\Pi$ such that $G_1 = \Pi(G_2)$, or more formally, if there is a permutation $\Pi$ such that $(i,j)$ is an edge in $G_1$ if and only if $(\Pi(i),\Pi(j))$ is an edge in $G_2$. The Graph Isomorphism Problem is the problem of deciding whether two given graphs are isomorphic.

Is there an operation on graphs that produces "gap amplification" in the style of Dinur's proof of the PCP Theorem? In other words, is there a polynomial time computable transformation from $(G_1,G_2)$ to $(G'_1,G'_2)$ such that 


if $G_1$ and $G_2$ are isomorphic, then $G'_1$ and $G'_2$ are also isomorphic, and
if $G_1$ and $G_2$ are not isomorphic, then for each permutation $\Pi$, the graph $G'_1$ is "$\epsilon$-far" from $\Pi(G'_2)$ for some small constant $\epsilon$, where $\epsilon$-far means that if we choose $(i,j)$ uniformly at random, then with probability $\epsilon$ either

$(i,j)$ is an edge of $G'_1$ and $(\Pi(i),\Pi(j))$ is not an edge of $G'_2$, or
$(i,j)$ is not an edge of $G'_1$ and $(\Pi(i),\Pi(j))$ is an edge of $G'_2$.
I do not know if such a thing could exist or not. But it is interesting (and perhaps timely) to note that such a "gap amplification" would likely imply a quasipolynomial time algorithm for graph isomorphism (different than the recently announced one)

In this paper, an approximation algorithm is given for the "MAX-PGI" problem of maximizing matched pairs of edges/non-edges; if we reduce from GI to "Gap-MAX-PGI", then we can approximate to distinguish which side of the gap we are on.

So, I think Dinur's proof of the PCP theorem is unlikely to be directly generalizable to such a "gap amplifier", given the hurdles that would have to be overcome.