Connectedness-locus embedding conjecture for exponential and error functions

Let $E_c(z)=\exp z+c$ be the exponential family, and let

$$\mathscr{E}_a(z)=a\int_{0}^{z}e^{-w^2}\,dw$$

for $a\in\mathbb{C}^*$. Define the bounded-orbit parameter sets

$$\mathcal{M}_{\exp}=\big\{c\in\mathbb{C}\mid \{E_c^{\circ n}(c)\}_{n\geq 0}\text{ is bounded}\big\}$$

and

$$\mathcal{M}_{\mathrm{err}}=\big\{a\in\mathbb{C}^*\mid \{\mathscr{E}_a^{\circ n}(a)\}_{n\geq 0}\text{ is bounded}\big\}.$$

Connectedness-locus embedding conjecture. There exists a continuous map from $\mathcal{M}_{\exp}$ to $\mathcal{M}_{\mathrm{err}}$ that is a homeomorphism onto its image.

This conjecture proposes an analogue, for the exponential and error-function families, of the structural relationship established in the paper between unicritical polynomial approximations and their limiting entire functions. The source does not provide a resolution, so the conjecture remains open.