Boundary-slope interval conjecture for exceptional surgeries

Let $K$ be a hyperbolic knot in $S^3$ that admits non-trivial exceptional surgeries. A NIT boundary slope is a boundary slope that is non-integral or toroidal. For $b e{}$ an integer, \left\lfloorb\right\rfloor and \left\lfloorb\right\rfloor are respectively the floor and ceiling of $b$; for integral $b$, both equal $b$. Boundary-slope interval conjecture. There are possibly equal NIT boundary slopes $b_1\leq b_2$ such that all exceptional surgeries occur as rational numbers in

$$[\lfloor b_1\rfloor,\lceil b_2\rceil],$$

or in the singleton set $\{\lfloor b_1\rfloor\}$ if $\lfloor b_1\rfloor=\lceil b_2\rceil$, and, if $\lceil b_1\rceil\leq\lfloor b_2\rfloor$, every integer in $[\lceil b_1\rceil,\lfloor b_2\rfloor]$, or the corresponding singleton when the endpoints agree, is exceptional. The conjecture is proposed as a refinement of the earlier boundary-slope and consecutive-integer conjectures, supported by enumerated exceptional fillings for many knot complements; its general status remains open.