Propagation of excess normal defect for cyclic matrices

Let $A \in \mathbb{C}^{n\times n}$ and $\tilde{A}\in \mathbb{C}^{(n+1)\times (n+1)}$ with $n \geq 4$ be the matrices defined by

$$A = \begin{pmatrix} 0 & a_1 & & \cdots & 0 \\ & & a_2 & & \\ \vdots & & & \ddots & \vdots \\ & & & & a_{n-1} \\ a_n & & & \cdots & 0 \end{pmatrix},\qquad \tilde{A} = \begin{pmatrix} 0 & a_1 & & \cdots & & 0 \\ & & a_2 & & \\ \vdots & & & \ddots & & \vdots \\ & & & & a_{n-1} & \\ & & & & & a_n \\ 0 & & & \cdots & & 0 \end{pmatrix}.$$

Here $\operatorname{nd}$ is the normal defect and $\varepsilon$ is the positive index of inertia of the commutator used in the paper. Propagation conjecture. If

$$\operatorname{nd}(A)=\varepsilon(A)+1,$$

then

$$\operatorname{nd}(\tilde{A})=\varepsilon(\tilde{A})+1.$$

The claim is motivated by an explicit example and by the apparent structural similarity between normal completions of $A$ and $\tilde{A}$; its general validity is not established.