The bench was first targeted last Tuesday (June 14), but scrubbed clean by three kind schoolboys. However, it has been confirmed that the seat has been defaced once again.
June told the LBO: "For me, this act of vandalism hurt personally because it was directed at one of our town’s biggest community champions.
"Wally was well known, loved and respected for the work he did in the community for over 60 decades or more.
"And it's hard to understand the mindset of these young people who think it’s OK to deface a memorial bench (and many more public areas of our town) and not be held accountable for their actions."
She added: "I believe that the money will appeal to the immoral morals of their friends who will give them up in an instant, so that the whole town can know who they are. I’m hoping that the lesson of public shaming will make them think twice in the future."
Cllr Carberry told the LBO: "The problem of the graffiti seems to be getting more prolific. There was a great deal of comment on social media relating to the defacing of the Wally Randall Memorial bench and the kind act by three young children who sought to clean up the damage.
"Since that story was published the bench has again been tagged with graffiti. Vandalism is unpleasant at anytime but it is a whole other thing when those who perpetrate such crime target memorials.
"I believe it is important to identify those responsible for it to stop which is why I am prepared to offer a reward out of my own pocket for information leading to the arrest and conviction of those responsible."
Since the appeal's launch, Cllr Carberry and June both confirmed that some sources have come forward, and that information is being prepared to hand over to the Leighton Buzzard community policing team.
However, they are still looking for additional names and information.
Cllr Carberry added: "As an elected representative of our community I have a duty to protect the environment we all live in and the cost of rectifying the vandalism comes from the public purse so we are all, in a way, victims.
"I am working with members of our community, the town council and our towns community police team to resolve the matter and for those committing these offences be aware, you are on our radar."
June said: "I am just waiting on that information before meeting with our local community policing team, which I am sure Cllr Carberry will also attend as he is very supportive of our community."
She added: "In defence of these young people, there is very limited provision in this town to keep them occupied that doesn’t involve paying for the activity. And once they have outgrown the Brownies and Scouts, where do they go?
"As a community, we need to direct these young people into meaningful activities that keep them occupied."
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