This is an old thread, but I figured I would add a short comment that may explain some of the reasoning behind the 'wishful thinking' that Neves refers to.

Let us, for all integer $p \geq 1$, write $\omega_p$ for the $p$-width of $\mathbf{S}^3$, that is the min-max value associated to a $p$-parameter Almgren-Pitts construction. Then
\begin{equation}
\omega_1 \leq \cdots \leq \omega_p \leq \cdots
\end{equation}
is an increasing, though not strictly increasing sequence.

Moreover, for each $p$ there is an embedded minimal surface $\Sigma_p \subset \mathbf{S}^3$ with area and Morse index satisfying
\begin{equation}
\mathrm{area} \, \Sigma_p = \omega_p
\quad \text{and} \quad \mathrm{index} \, \Sigma_p \leq p.
\end{equation}
Note that the index bound is not sharp in general. For example $\omega_1 = \cdots = \omega_4 = 4 \pi$ are all realised by an equatorial sphere, and therefore
\begin{equation}
\mathrm{index} \, \Sigma_4 = 1.
\end{equation}

The specific values of the widths are only known for relatively small $p$, but a PhD student of Neves, Charles Nurser, proved some estimates for the ninth width, namely:
\begin{equation}
2 \pi^2 < \omega_9 < 8 \pi.
\end{equation}

The lower bound means that $\Sigma_9$ is not a Clifford torus, and the upper bound that it is not a multiplicity-two copy of a minimal sphere. It must therefore be a 'new' surface, and it is not unreasonable to conjecture that it might be the Lawson surface $\xi_{1,2}$.

This is just a guess, but as far as the Morse index is concerned, perhaps the thinking was that $\omega_p > \omega_{p-1}$ might mean that $\mathrm{index} \, \Sigma_p = p$. For example $\omega_5 = 2 \pi^2 > 4 \pi = \omega_4$, and is realised by a Clifford torus, with Morse index five. However I must emphasise this is purely speculative, and I'd be very curious to hear from someone more knowledgeable.