The trace-norm Hlawka constant is infinite

For a norm, let $N$ be the three-summand defect
$\sum_j\|X_j\|-\|X_1+X_2+X_3\|$ and $D$ the sum of the three pairwise defects
$\|X_i\|+\|X_j\|-\|X_i+X_j\|$. For the trace norm (Schatten $p=1$), no
finite constant $C_1$ gives $N\le C_1D$. The proof is an explicit family:
three unit vectors $u_j\in\mathbb R^2$ at mutual angles of $120^\circ$, lifted
to $v_j=(\sqrt{1-t}\,u_j,\sqrt t)\in\mathbb R^3$, give rank-one matrices
$X_j=u_jv_j^{\mathsf T}$; for each pair $i\ne j$ the Gram matrices of $(u_i,u_j)$
and of $(v_i,v_j)$ commute, so every singular value is explicit. The defect ratio equals
$(1-\sqrt{1-t})/(2-q(t))$ with $q(t)=\tfrac12\sqrt{1+3t}+\tfrac32\sqrt{1-t}$,
which is asymptotic to $2/(3t)$ as $t\downarrow0$. The failure occurs already
for real $2\times3$ matrices, and for real $3\times3$ matrices after
appending a zero row. The constants $C_p$ for $1<p<\infty$ are not determined.