  For a finite quiver without sinks, we establish an isomorphism in the
homotopy category $\mathrm {Ho}(B_\infty)$ of $B_{\infty}$-algebras between the
Hochschild cochain complex of the Leavitt path algebra $L$ and the singular
Hochschild cochain complex of the corresponding radical square zero algebra
$\Lambda$. Combining this isomorphism with a description of the dg singularity
category of $\Lambda$ in terms of the dg perfect derived category of $L$, we
verify Keller's conjecture for the singular Hochschild cohomology of $\Lambda$.
More precisely, we prove that there is an isomorphism in
$\mathrm{Ho}(B_\infty)$ between the singular Hochschild cochain complex of
$\Lambda$ and the Hochschild cochain complex of the dg singularity category of
$\Lambda$. One ingredient of the proof is the following duality theorem on
$B_\infty$-algebras: for any $B_\infty$-algebra, there is a natural
$B_\infty$-isomorphism between its opposite $B_\infty$-algebra and its
transpose $B_\infty$-algebra.
  We prove that Keller's conjecture is invariant under one-point (co)extensions
and singular equivalences with levels. Consequently, Keller's conjecture holds
for those algebras obtained inductively from $\Lambda$ by one-point
(co)extensions and singular equivalences with levels. These algebras include
all finite dimensional gentle algebras.
