Doesn't matter on the infallibility or not of a lie detector. It was the woman who said she would take one and offered it then withdrew her offer with no further comment.
As or the UK picking up the bill for investigation then that is bonkers. Note the UK police are not investigating any crimes alleged to have happened in Sweden are they despite the 40k a day police bill.
Giving this woman some form of special treatment when she could easily fly back to Portugal and ask the Portuguese to continue the investigation would be far better.
After all the UK police could not find a body in the same house that they had been in 5 times in one week.
It seems that your opinion is only allowed in the press or public on this matter if your mind is partly closed and definitely not open to the prospect that perhaps one of the family or friends were involved, Whether in covering up a tragic accident or otherwise. ALL avenues should be investigated and as so often it seems to be in cases like this, the truth lies very close to home.
After all, after this length of time she should be fluent in Portuguese and very au fait with their legal system though reading anything she can about it in order to ensure she gets her rights protected and exercised.