= Solution
Only a shell of <energy> width of order $k_BT$ near the <Fermi energy> can change occupation: much lower states are blocked by the <Pauli exclusion principle>. The <thermal-shell estimate of electronic heat capacity> counts
$$
N_{\rm act}\sim g(E_F)k_BT=\frac32N\frac{T}{T_F}.
$$
To recover the rough coefficient requested in the paper, assign each of these active particles an excitation <energy> of order $3k_BT/2$. Then
$$
\Delta U\sim\frac32N_{\rm act}k_BT=\frac94Nk_B\frac{T^2}{T_F},\qquad
\boxed{C_V^{\rm el}\sim\frac92Nk_B\frac{T}{T_F}}.
$$
The differentiation matters: the active shell itself widens as <temperature> rises. This argument explains the small linear-in-$T$ <electronic heat capacity>; its shell width and average excitation <energy> are heuristic, so it does not establish a precise numerical prefactor.
A controlled calculation from the <Fermi-Dirac distribution> gives a nearby but different result. At fixed particle number, the <Sommerfeld expansion> yields
$$
U(T)=\frac35NE_F+\frac{\pi^2}{4}N\frac{(k_BT)^2}{E_F}+O(T^4),\qquad
\boxed{C_V^{\rm el}=\frac{\pi^2}{2}Nk_B\frac{T}{T_F}+O((T/T_F)^3Nk_B)}.
$$
Thus $9/2$ is a rough estimate of the exact free-electron coefficient $\pi^2/2\simeq4.935$, not an exact consequence of the stated <density of states>. Both require $T\ll T_F$.
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