Let $f:[0,1]^2 \rightarrow \mathbb C$ be an $H^1$ function with the property that $f(x,x)=0$ and $\Vert f \Vert_{L^2[0,1]}=1.$

Does there exist a constant $c>0$ such that any such function satisfies

$$ \Vert f-1 \Vert_{H^1}>c?$$

I was thinking that the Fourier series could help to prove or disprove something like this, but I did not get far so far.

It would be clearly possible in $L^2$ norm let's say, but I find it tricky in Sobolev norms.