For each real number $n$ > 6, we prove that there is a sequence ${pk(n,z)}^{infty}_{k=1}$ of fourth degree self-reciprocal polynomials such that the zeros of $p_k(n,z)$ are all simple and real, and every $p_{k+1}(n,z)$ has the largest (in modulus) zero ${alpha}{eta}$ where ${alpha}$ and ${eta}$ are the first and the second largest (in modulus) zeros of $p_k(n,z)$, respectively. One such sequence is given by $p_k(n,z)$ so that $$p_k(n,z)=z^4-q_{k-1}(n)z^3+(q_k(n)+2)z^2-q_{k-1}(n)z+1$$, where $q_0(n)=1$ and other $q_k(n)^{prime}s$ are polynomials in n defined by the severely nonlinear recurrence $$4q_{2m-1}(n)=q^2_{2m-2}(n)-(4n+1)prod_{j=0}^{m-2};q^2_{2j}(n),\4q_{2m}(n)=q^2_{2m-1}(n)-(n-2)(n-6)prod_{j=0}^{m-2};q^2_{2j+1}(n)$$ for $m{geq}1$, with the usual empty product conventions, i.e., ${prod}_{j=0}^{-1};b_j=1$.