The horizontal-or-vertical-line conjecture for $e^{-x^2-y^4}$

Let $\mathbb{R}^2$ carry the density

$$f(x,y)=\exp(-x^2-y^4).$$

An isoperimetric region minimizes weighted perimeter among regions of prescribed weighted volume. Horizontal-or-vertical-line conjecture. Every isoperimetric region is bounded by a horizontal line for prescribed volumes close to half the total volume, or by a vertical line. The density has finite total mass, so isoperimetric regions exist for every prescribed volume; the source presents this as a conjecture motivated by known results for related Gaussian densities and gives no resolution.