Thank you for that.
Provincial and territorial governments are responsible for forest management and they specify an annual allowable cut from public lands. Regulating harvest levels in this way helps to ensure the sustainability over the long term, regardless of the changes in demand for wood. So the short answer to your question would be “no”.
Timber harvesting is sustainable in Canada. There are strong laws, oversight in management and the requirement for all harvested public lands to be regenerated. According to the National Forestry Database, in 2020, the sustainable wood supply in Canada was calculated at 215 million cubic metres, of which 143.1 million cubic metres were harvested. That's 66.5%, so in general I'd say no.