Bang's half-area cylinder covering question (and the 1-codimensional cylinder covering conjecture)

Theorem 1.1: a regular tetrahedron $K$ with $A_{\min}(K)=\sqrt2$ has, for each $0<\varepsilon\le1/2000$, a cover by $2\lceil2/\varepsilon^2\rceil$ cylinders with compact triangular perpendicular bases and $\frac1{\sqrt2}\sum|B_i|=\frac12-\frac{13}{6000}\varepsilon^2+O(\varepsilon^4)$, strictly below half. Corollary 1.2: every nondegenerate tetrahedron has a finite cover with directionwise relative cost below 1/2, refuting the Bezdek-Khan conjecture. Not shown: the optimal constant (Bezdek-Litvak's lower bound 1/3 stands) or counterexamples among centrally symmetric bodies.

Theorem 1.1: a regular tetrahedron $K$ with $A_{\min}(K)=\sqrt2$ has, for each $0<\varepsilon\le1/2000$, a cover by $2\lceil2/\varepsilon^2\rceil$ cylinders with compact triangular perpendicular bases and $\frac1{\sqrt2}\sum|B_i|=\frac12-\frac{13}{6000}\varepsilon^2+O(\varepsilon^4)$, strictly below half. Corollary 1.2: every nondegenerate tetrahedron has a finite cover with directionwise relative cost below 1/2, refuting the Bezdek-Khan conjecture. Not shown: the optimal constant (Bezdek-Litvak's lower bound 1/3 stands) or counterexamples among centrally symmetric bodies.