= Solution
For the anticlockwise boundary of the region, the inward normal is the left normal, so the <signed curvature> obeys the <turning tangent theorem> $\int_0^L k(s)\,ds=2\pi$. Therefore some $s_0$ has $k(s_0)\leq2\pi/L$. The enclosed area satisfies $A\geq d_1d_2$, and the <planar isoperimetric inequality> gives $L\geq2\sqrt{\pi A}$. Consequently
$$
\boxed{k(s_0)\leq\frac{2\pi}{L}\leq\sqrt{\frac{\pi}{A}}\leq\sqrt{\frac{\pi}{d_1d_2}}}.
$$
This uses the interpretation of the curve as a simple boundary traversed once; the inward normal and enclosed region in the question have that meaning.
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