A counterexample to the $$\phi $$ ϕ -dimension conjecture

In 2005, the second author and Todorov introduced an upper bound on the finitistic dimension of an Artin algebra, now known as the $$\phi $$ -dimension. The $$\phi $$ -dimension conjecture states that this upper bound is always finite, a fact that would imply the finitistic dimension conjecture. In this paper, we present a counterexample to the $$\phi $$ -dimension conjecture and explain where it comes from. We also discuss implications for further research and the finitistic dimension conjecture.