Malleability characterization of exponentiable virtual double categories

Let $\mathbb{X}$ be a virtual double category. Say that $\mathbb{X}$ is exponentiable when the exponential by $\mathbb{X}$ exists, and say that it is malleable when its multiplication cell, viewed as a monoid for the relevant monad, is cartesian.

Malleability conjecture. The following conditions are equivalent:

1. $\mathbb{X}$ is exponentiable.
2. $\mathbb{X}$ is malleable.
3. $\mathbb{X}$ satisfies a restricted form of malleability expressing every cell as a composite of nullary, unary, and binary cells.
4. The exponential $\mathsf{Span}^{\mathbb{X}}$ exists.

This conjecture aims to characterize exponentiability of virtual double categories by a decomposition property for multiary cells, generalizing the corresponding characterization for multicategories. The equivalence of conditions 1 and 4 has been verified, while the remaining equivalences are not established here; condition 3 is intentionally left without a precise formulation.