Consider the special multiatomic Sobolev orthogonal polynomials of Ex. 3.8 with
$$
K=L=c_1^0=c_1^1=1 \quad \text { and } \quad \mathrm{d} \lambda(t)=\mathrm{d} t \text { on }[-1,1]
$$
(c.. [2, Proposition 3.2]). Test the following conjectures.
(a) Conjecture 1: for $m 0=m_1^0 \geq 0, m 1=m_1^1>0$ and for all $n \geq 3$, the zeros of $\pi_n$ are all in the interior of $[-1,1]$ except for one, which is larger than 1.
(b) Conjecture 2: The exceptional zeros $z_n>1$ of $\pi_n$ tend monotonically to 1 for $n \geq 4$,
$$
z_4>z_5>\cdots>1, \quad \lim _{n \rightarrow \infty} z_n=1
$$
(c) Conjecture 3: The zeros of $\pi_n$ in the interior of $[-1,1]$ are all simple zeros.
(d) Investigate the interlacing of the zeros of the Sobolev polynomials $\pi_k(\cdot ; S)$ among themselves and with the zeros of the orthogonal polynomials $\pi_k(\cdot ; \mathrm{d} \lambda)$. Specifically, test the following conjectures.
Conjecture 4: If $n=2$, the zero of $\pi_1(\cdot ; S)$ interlaces with the two zeros of $\pi_2(\cdot ; S)$; if $n>2$, only the first $n-2$ zeros of $\pi_{n-1}(\cdot ; S)$ interlace with those of $\pi_n(\cdot ; S)$.
Conjecture 5: If $n=2$, the zero of $\pi_1(\cdot ; \mathrm{d} \lambda)$ interlaces with the two zeros of $\pi_2(\cdot ; S)$; if $n>2$, only the first $n-2$ zeros of $\pi_{n-1}(\cdot ; \mathrm{d} \lambda)$ interlace with those of $\pi_n(\cdot ; S)$.
The statements for $n>2$ in Conjectures 4 and 5 is all that can be expected, given the monotonicity property in Conjecture 2 for the exceptional zeros.