Mode-by-mode Morse index conjecture for the critical spherical catenoid

Let $\Sigma_a$ be the critical spherical catenoid with parameter $a>1/2$. Decompose its Robin Morse index into Fourier modes $|k|=0$, $|k|=1$, and $|k|\geq 2$. Mode-by-mode index conjecture. For every $a>1/2$, the Robin Morse index of $\Sigma_a$ in mode $|k|=0$ equals $2$, and its Robin Morse index in modes $|k|\geq 2$ equals $0$. Consequently, together with the established contribution of mode $|k|=1$, one would have

$$\operatorname{ind}(\Sigma_a)=4,\qquad \operatorname{nul}(\Sigma_a)=\operatorname{nul}(\Sigma_a)|_{|k|=1}=2.$$

The conjecture would settle Medvedev's conjecture on the Robin Morse index and nullity of the critical spherical catenoid. The mode $|k|=1$ contribution is known to have index exactly $2$, while the asserted contributions from modes $|k|=0$ and $|k|\geq 2$ remain to be proved.