The nil-blob algebra: An incarnation of type ${\tilde{A}}_1$ Soergel calculus and of the truncated blob algebra

We introduce a type $B$ analogue of the nil Temperley-Lieb algebra in terms of generators and relations, that we call the (extended) nil-blob algebra. We show that this algebra is isomorphic to the endomorphism algebra of a Bott-Samelson bimodule in type ${\tilde{A}}_1$. We also prove that it is isomorphic to an idempotent truncation of the classical blob algebra.Thus we provide strong evidence in favor of the recent Blob vs. Soergel conjecture.