 Solution

Let $b\simeq 2l_p$ be the Kuhn length, $N=L/b$ the number of statistical segments, and $v$ their effective excluded volume. We first assume a good-solvent, strongly confined self-avoiding walk, rather than an ideal chain. A confinement blob contains $g$ links and has diameter $D\sim b{g}^{3/5}$ in the simple $v\sim b^3$ scaling model. Therefore $g\sim (D/b{)}^{5/3}$. Distant blobs cannot freely pass through one another in a narrow tube, so the $N/g$ blobs form an axial string. The de Gennes confinement scaling is consequentlyThe free energy is of order $k_{B}T$ per blob, not the energy of stretching a straight elastic rod.

The role of solvent and chain thickness can be made explicit by a Flory approximation: an axial extension $X$ costs an entropic term $k_{B}T{X}^2/(N{b}^2)$, while the confined number density gives an interaction term $k_{B}Tv{N}^2/(X{D}^2)$. Minimizing their sum gives $X^3\sim v{N}^3{b}^2/D^2$, henceFor a thin statistical segment of effective diameter $d$, $v\sim b^{2}d$ instead gives $X\sim L(bd/D^2{)}^{1/3}$. Thus the simpler answer implicitly fixes the excluded volume; the DNA thickness and ionic environment can matter quantitatively.

Both blob arguments require $b\ll D\ll R_{\mathrm{free}}$ and $N/g\gg 1$, where $R_{\mathrm{free}}\sim b{N}^{3/5}$ in the simple good-solvent model. The printed comparison with $L$ alone does not ensure confinement relative to the unconfined coil. If $D\gg R_{\mathrm{free}}$, the tube barely perturbs the coil. An ideal chain without excluded volume has axial spread of order $\sqrt{Lb}$ and does not acquire the extensive extension of a nonoverlapping blob string. These are distinct physical regimes, not changes to a numerical prefactor.