Invariance of pedal unfolding and Legendrian duality under A-equivalence

Let $\varphi_1,\varphi_2:(\mathbb{R}^2,0)\to(\mathbb{R}^3,0)$ be two $C^\infty$ map-germs of pedal unfolding type. Let $\Phi_1,\Phi_2:(\mathbb{R}^2,0)\to(\mathbb{R}^3,0)$ be two normalized Legendrian map-germs. Invariance conjecture. If $\varphi_1$ is $\mathcal{A}$-equivalent to $\varphi_2$, then $\mathcal{I}(\varphi_1)$ is $\mathcal{A}$-equivalent to $\mathcal{I}(\varphi_2)$; and if $\Phi_1$ is $\mathcal{A}$-equivalent to $\Phi_2$, then $\mathcal{D}(\Phi_1)$ is $\mathcal{A}$-equivalent to $\mathcal{D}(\Phi_2)$. This conjecture naturally extends the preceding bijectivity theorems by asserting that the constructions also respect $\mathcal{A}$-equivalence. The supplied text gives no resolution of either assertion.