It can, where the supervision is the design professional's oversight of the work against the drawings and specifications. It does not where the applicant is effectively managing the contractor's operations, scheduling and workforce, which describes construction management rather than landscape architecture.
Name the activity precisely in the documents
The word supervision on its own invites the wrong reading. Ask the client or employer to write what actually happens on site: periodic observation visits, review of shop drawings, responses to requests for information, punch lists and certification for payment. If the honest description is running the build, discuss with counsel whether the assignment can be restructured or whether a different approach is needed.
Hypothetical example: a Canadian irrigation designer is offered a role covering both system design and direction of the installation crew. Irrigation designer is a job title rather than an entry on the USMCA professions list, so a TN request would turn on whether the duties fit a profession that does appear on that list and whether the applicant holds that profession's prescribed qualification. Neither point is settled by the title written on the offer.
Where the duties are largely design and specification, ask counsel which listed entry the employer believes applies and what that entry requires; where the duties are largely crew direction, the question is a different one. Write the duty list first and choose the description afterwards, not the other way round.