Modified Integer Round-Up Property for the cutting stock problem

Let $P_{CSP}$ be the set of integer cutting patterns for the cutting stock problem, let $a_{ip}$ denote the number of copies of item $i$ in pattern $p$, and let $d_i$ be the demand for item $i$. The cutting stock problem is the integer set-covering model

$$\min \sum_{p\in P_{CSP}}\lambda_p$$

subject to

$$\sum_{p\in P_{CSP}}a_{ip}\lambda_p\geq d_i \quad\text{for all }i\in I,\qquad \lambda_p\in\mathbb{Z}_+ \quad\text{for all }p\in P_{CSP},$$

and its proper relaxation is obtained by relaxing the integrality constraints on the pattern variables. Modified Integer Round-Up Property (MIRUP). The difference between the optimal solution value of the cutting stock problem and the rounded-up optimal solution value of its proper linear relaxation is at most one. This property concerns the strength of the proper relaxation: it asserts that rounding its optimum up gives a value within one of the integer optimum, but the source does not state whether the property has been proved or disproved.