Conjecture that the higher Newton polygon family is a $p$-integral basis

Let $p$ be a prime, let $K=\mathbb Q(\theta)$ be the number field generated by a root $\theta$ of a monic irreducible polynomial, and let $\operatorname{CT}$ be the set of complete types used by the higher Newton polygon algorithm. For each regular reduced type $\mathbf t^0$, let $\mathcal Q_{\mathbf t^0}$ be the family constructed from the corresponding quotients; for each complete type $\mathbf t$ whose reduced type is not regular, let $\mathcal Q_{\mathbf t}$ be the family constructed using the additional quotient $q(\theta)$ and the associated denominator. Define

$$\mathcal B=\left(\bigcup_{\mathbf t^0\in\operatorname{CT}^{\operatorname{reg}}}\mathcal Q_{\mathbf t^0}\right)\cup\left(\bigcup_{\mathbf t\in\operatorname{CT},\,\mathbf t^0\notin\operatorname{CT}^{\operatorname{reg}}}\mathcal Q_{\mathbf t}\right).$$

Higher Newton polygon basis conjecture. The following family is a $p$-integral basis of $K$:

$$\mathcal B=\left(\bigcup_{\mathbf t^0\in\operatorname{CT}^{\operatorname{reg}}}\mathcal Q_{\mathbf t^0}\right)\cup\left(\bigcup_{\mathbf t\in\operatorname{CT},\,\mathbf t^0\notin\operatorname{CT}^{\operatorname{reg}}}\mathcal Q_{\mathbf t}\right).$$

This is the assembled basis claim produced by the algorithm in its regular and nonregular cases. In the supplied text it is stated as a conjectural family being a $p$-integral basis, and no resolution is given.