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We prove the claim. Put $r=3^{-1/4}$ and $p(z)=z^4+4r^3z$. The three critical points are

$$c_j=r\exp((2j+1)\pi i/3),\qquad j=0,1,2.$$

They satisfy $c_j^3=-r^3$, so $p(c_j)=3r^3c_j$ and $|p(c_j)|=3r^4=1$. For $0\le s<1$,

$$p(sc_j)=\frac{4s-s^4}{3}p(c_j),\qquad 0\le\frac{4s-s^4}{3}<1.$$

The segment from zero to each $c_j$, excluding its endpoint, therefore lies in the same component $\Omega$ of $\{|p|<1\}$. All three $c_j$ belong to $\partial\Omega$.

There are no critical points inside $\Omega$. The restriction of $p$ to that component is a proper unramified map to the unit disc, hence a conformal bijection: an unramified connected cover of the simply connected disc has one sheet. Its inverse $\phi$ extends continuously to the closed disc. Indeed an inverse branch of a polynomial has a convergent Puiseux expansion at each boundary critical value, and is analytic at the remaining boundary values. These expansions also make the boundary curve rectifiable. The radial identities above give $\phi(p(c_j))=c_j$. Traversing the unit circle visits the three values $p(c_j)$ once each; applying the triangle inequality to the three arcs of the closed curve $\phi(\mathbf T)$ gives

$$\operatorname{length}(\partial\Omega) \ge |c_0-c_1|+|c_1-c_2|+|c_2-c_0| =3\sqrt3r=3^{5/4}.$$

For completeness, the boundary parameterization counts each regular boundary arc once. Two distinct arguments cannot have the same boundary image, since $p\circ\phi$ is the identity on the closed disc. Thus its curve length is the boundary length used above.

Write $\ell_0=\ell(z^4-1)$. Solving for the four inverse roots on the unit circle gives the density $|1+e^{i\theta}|^{-3/4}$, and therefore

$$\frac{\ell_0}{4} =2^{-3/4}\int_0^{\pi/2}(\sin u)^{-3/4}\,du \le2^{-3/4}\int_0^{\pi/2}(2u/\pi)^{-3/4}\,du =2^{1/4}\pi<3^{5/4}.$$

The last strict inequality follows, for example, from $\pi<22/7$. Combining the two estimates proves the claim. The other petals can be shorter; this argument concerns a proposed bound for each petal individually and does not refute a bound on their total length.