In large-scale distributed storage systems, erasure codes are used to achieve fault tolerance in the face of node failures. Tuning code parameters to observed failure rates has been shown to significantly reduce storage cost. Such tuning of redundancy requires "code conversion", i.e., a change in code dimension and length on already encoded data. Convertible codes are a new class of codes designed to perform such conversions efficiently. The access cost of conversion is the number of nodes accessed during conversion. Existing literature has characterized the access cost of conversion of linear MDS convertible codes only for a specific and small subset of parameters. In this paper, we present lower bounds on the access cost of conversion of linear MDS codes for all valid parameters. Furthermore, we show that these lower bounds are tight by presenting an explicit construction for access-optimal linear MDS convertible codes for all valid parameters. En route, we show that, one of the degrees-of-freedom in the design of convertible codes that was inconsequential in the previously studied parameter regimes, turns out to be crucial when going beyond these regimes and adds to the challenge in the analysis and code construction.