  An on-line chain partitioning algorithm receives a poset, one element at a
time, and irrevocably assigns the element to one of the chains in the
partition. The on-line chain partitioning problem involves finding the minimal
number of chains needed by an optimal on-line algorithm. Chrobak and \'Slusarek
considered variants of the on-line chain partitioning problem in which the
elements are presented as intervals and intersecting intervals are
incomparable. They constructed an on-line algorithm which uses at most $3w-2$
chains, where $w$ is the width of the interval order, and showed that this
algorithm is optimal. They also considered the problem restricted to intervals
of unit-length and while they showed that first-fit needs at most $2w-1$
chains, over $30$ years later, it remains unknown whether a more optimal
algorithm exists. In this paper, we improve upon previously known bounds and
show that any on-line algorithm can be forced to use $\lceil\frac{3}{2}w\rceil$
chains to partition a semi-order presented in the form of its unit-interval
representation. As a consequence, we completely solve the problem for $w=3$.
