By the AM-GM inequality,
$$\begin{aligned} &abc + bcd + dca + dab \ge 4\sqrt[4]{(abcd)^3} \quad \text{and} \\ &ab + bc + cd + da + ac + bd \ge 6\sqrt[6]{(abcd)^3}, \end{aligned}$$
with equality in both inequalities iff $a = b = c = d$. Hence, letting $x = \sqrt[4]{abcd}$, we have that
$$1 \ge 3x^4 + 8x^3 + 6x^2$$
with equality iff $a = b = c = d$. But $1 - 3x^4 - 8x^3 - 6x^2 = (1 - 3x)(1+x)^3$, and $x$ is positive. Hence, $1 \ge 3x^4 + 8x^3 + 6x^2$ iff $x \le 1/3$, with equality iff $x = 1/3$. Thus $abcd \le 1/3^4$, and the inequality is strict unless $a = b = c = d = 1/3$.