Regardless, if the employer and the employee haven't reached a resolution of the issue, the employer is obligated to have a competent person and is obligated to pay for it. The only time the labour program would come in is if they don't do that.
The employer is obligated to conduct the investigation, and if they hire a third party to do it, then the labour program would only come and investigate if that process hadn't been followed, so the costs for this competent person would be borne by the employer.