Liberal and conservative, as with any descriptive terminology is subjective; generally, without a person experiencing both of them first hand, any answer can be inaccurate and best, and just plain fallacy at worst.
Conservative can be defined as traditional, respecting 'the old ways'.
Liberal can be defined as progressive, looking to the present and thinking forward to the future.
In teaching, basing one’s entire pedagogy, irrespective of one, or the other, means a disaster in regards to learning; it creates a paradox in logic, and an oxymoron to learning.
For example:
To be an effective teacher, the most basic skill at our disposal is reflection; “looking to the past to adapt to a future”. Taking a solely conservative approach means, we disregard that times change, and new needs have to be met. Taking a solely liberal approach, would mean, all tradition is redundant and obsolete because new is better; rhetorically speaking, things didn't just manifest in the present (how did we arrive here? ).
Speaking from a practical-piano standpoint, these two ideologies are, in actuality, referring to how one is taught to play the instrument.
A conservative approach might adopt the teaching of scales and exercises, focussing on reading music from sheet manuscript, also learning traditional harmony; a conservatory approach if you will.
Whereas, a more-liberal approaches might omit learning to read entirely, using tablet and phone apps or YouTube videos. The key difference being, freedom in approach different to the preordained methodologies.
My opinion on this topic is why not both? The ends dictates the means.
Eventually everyone uses a pencil in the traditional way; yes, it might be quicker to take shortcuts to the prize, but in order to recreate it, we must use a tried and tested method, which ultimately is, or will be, the traditional one.