Holland's coefficient-energy constant: determine $\Lambda_n$ and the limit $\Lambda=\lim\Lambda_n/n$ for polynomials of positive real part

Let $\mathcal{P}_n$ be the class of polynomials $p(z)=1+a_1z+\dots+a_nz^n$ of degree at most $n$ with $p(0)=1$ and $\operatorname{Re}p(z)>0$ throughout the open unit disc $D$. Holland's problem asks for the maximal $L^2$ energy of such $p$ on the unit circle, $\max_{p\in\mathcal{P}_n}\int_0^{2\pi}|p(e^{i\theta})|^2\,d\theta$. By Parseval's identity this equals $2\pi\,\Lambda_n$ where $\Lambda_n=\max_{p\in\mathcal{P}_n}\sum_{\nu=0}^{n}|a_\nu|^2$ (with $a_0=1$). There are two deliverables: (i) determine the exact value of $\Lambda_n$ for values of $n$ beyond those already computed; (ii) determine the limit $\Lambda=\lim_{n\to\infty}\Lambda_n/n$ (conjectured to exist), currently known only to satisfy $\tfrac23\le\Lambda\le1$.

Acceptance. ADVANCES: the exact value of $\Lambda_n$ (a rational or algebraic number, with the extremal polynomial and a certified SDP / exact-arithmetic verification) for one or more $n\ge3$ not previously known; each new $\Lambda_n$ is an independent, machine-checkable increment. ADVANCES: a rigorous improvement of the bounds $\tfrac23\le\Lambda\le1$, or a proof that the limit $\Lambda=\lim\Lambda_n/n$ exists. FULLY RESOLVES: the exact value of $\Lambda=\lim_{n\to\infty}\Lambda_n/n$ with proof. Floating-point estimates without exact or certified bounds do NOT qualify.